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nLab > Latest Changes: group extension

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeSep 12th 2010
    • (edited Oct 1st 2012)

    added to group extension a section on how group extensions are torsors and on how they are deloopings of principal 2-bundles, see group extension – torsors

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeOct 1st 2012
    • (edited Oct 1st 2012)

    am working on group extension. Have

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeOct 1st 2012
    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeOct 9th 2012

    Started a section Split exensions and semidirect product groups.

    Wanted to do much more. But the lab is mostly down, so it’s tedious. Have to quit now.

    • CommentRowNumber5.
    • CommentAuthorTim_Porter
    • CommentTimeOct 9th 2012
    • (edited Oct 9th 2012)

    Did you mean ‘an fiber sequence’ in the ideas section? I tried a fix but could not be sure that I had captured what you might have intended so undid it. (Fixed a typo later on). I found the difficulty was to capture the use of the lax action of GG on AA in simple enough terms for an ideas section. The use of fiber sequence does sort of do that, but then it was not clear to me that the entry on fiber sequences really was specific enough on the case of groups…. so I left it alone! In any case it is very nicely explained further down the entry.

    There seemed to be some hats missing. I think I fixed them all but may have missed some.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeOct 9th 2012
    • (edited Oct 9th 2012)

    Ah, I didn’t look at that first sentence in the Idea-section for a while…

    The statement of fiber sequence there is okay, but it should in addition say that the following equivalent statements hold (which in this form also work for ∞\infty-groups)

    • G≃G^⫽KG \simeq \hat G \sslash K;

    • the fiber sequence exhibits a normal inclusion.

    I’ll edit this as soon as the you have unlocked the entry.

    • CommentRowNumber7.
    • CommentAuthorTim_Porter
    • CommentTimeOct 9th 2012

    I have added a paragraph pointing out the link between the splittings of a split extension and non-abelian 1-cocycles / derivations

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeOct 9th 2012
    • (edited Oct 9th 2012)

    at group extension in the section Central extensions classified by group cohomology there was a two places silently assumed that the given 2-cocycle is normalized. I have now made this explicit.

    • CommentRowNumber9.
    • CommentAuthorTim_Porter
    • CommentTimeNov 7th 2013

    I have been trying to find universal central extension in the Lab (as there was a non-active link on another page). Has anyone seen an entry that handles this and its link with other areas?

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeNov 7th 2013

    There is this notion here: universal central extension

    But this needs disambiguation, one also speaks of “universal central extensions” for loop groups etc.

    • CommentRowNumber11.
    • CommentAuthorTim_Porter
    • CommentTimeNov 7th 2013

    Thanks. When I have time I will add in something on Milnor’s K-theory.

  12. Fix definition of cocycle from group extension

    Tobias Diez

    diff, v58, current

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeAug 23rd 2023

    added pointer to:

    diff, v59, current

1 to 13 of 13

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